I briefly skimmed the paper and don't see how you are getting this impression. Confidence intervals are—if we force the dichotomy—considered a frequentist rather than Bayesian tool. They point out that others are trying to squish a Bayesian interpretation on a frequentist tool by treating confidence intervals as though they are credible intervals, and they state this quite explicitly (p.17–18, emphasis mine):

Finally, we believe that in science, the meaning of our inferences are important. Bayesian credible intervals support an interpretation of probability in terms of plausibility, thanks to the explicit use of a prior. Confidence intervals, on the other hand, are based on a philosophy that does not allow inferences about plausibility, and does not utilize prior information. Using confidence intervals as if they were credible intervals is an attempt to smuggle Bayesian meaning into frequentist statistics, without proper consideration of a prior. As they say, there is no such thing as a free lunch; one must choose. We suspect that researchers, given the choice, would rather specify priors and get the benefits that come from Bayesian theory. We should not pretend, however, that the choice need not be made. Confidence interval theory and Bayesian theory are not interchangeable, and should not be treated as so.